An adjoint-based shape optimization approach for supersonic turbine cascades is proposed for application to organic Rankine cycle (ORC) turbines. The algorithm is based on an inviscid discrete adjoint method and encompasses a fast look-up table (LuT) approach to accurately deal with real-gas flows. The turbine geometry is defined by adopting state-of-the-art parameterization techniques (NURBS), enabling to handle both global and local control of the shape of interest. A preconditioned steepest descent method has been chosen as gradient-based optimization algorithm to efficiently search for the nearest minimum. The potential of the optimization approach is first verified by application on the redesign of an existing converging–diverging turbine nozzle operating in thermodynamic regions characterized by relevant real-gas effects. A significant efficiency improvement and a more uniform flow at the blade outlet section are achieved, with expected beneficial effects on the aerodynamics of the downstream rotor. The optimized configuration is also assessed by means of high-fidelity turbulent simulations, which point out the capability of the present inviscid approach in optimizing supersonic turbine cascades with very limited computational burdens. Finally, the newly developed real-gas adjoint method is compared against adjoints based on ideal equations of state on the same design problem. Results show that the performance gain obtained by a fully real-gas optimization strategy is by far higher than that achieved with simplified approaches in case of ORC turbines. This proves the relevance of including accurate thermodynamic models in all steps of ORC turbine design.

References

References
1.
Pini
,
M.
,
Persico
,
G.
,
Casati
,
E.
, and
Dossena
,
V.
,
2013
, “
Preliminary Design of a Centrifugal Turbine for ORC Applications
,”
ASME J. Eng. Gas Turbines Power
,
135
(
4
), p.
042312
.10.1115/1.4023122
2.
Petrovic
,
M. V.
,
Dulikravich
,
G. S.
, and
Martin
,
T. J.
,
2001
, “
Optimization of Multistage Turbines Using a Through-Flow Code
,”
J. Power Energy
,
215
(
5
), pp.
559
569
.10.1243/0957650011538802
3.
Larocca
,
F.
,
2008
, “
Multiple Objective Optimization and Inverse Design of Axial Turbomachinery Blade
,”
J. Propul. Power
,
24
(
5
), pp.
1093
1099
.10.2514/1.33894
4.
Pasquale
,
D.
,
Persico
,
G.
, and
Rebay
,
S.
,
2014
, “
Optimization of Turbomachinery Flow Surfaces Applying a CFD-Based Troughflow Method
,”
ASME J. Turbomach.
,
136
(
3
), p.
031013
.10.1115/1.4024694
5.
Pierret
,
S.
, and
Van Den Braembussche
,
R.
,
1999
, “
Turbomachinery Blade Design Using a Navier–Stokes Solver and Artificial Neural Network
,”
ASME J. Turbomach.
,
121
(
2
), pp.
326
332
.10.1115/1.2841318
6.
Rai
,
M.
,
2000
, “
Aerodynamic Design Using Neural Networks
,”
AIAA J.
,
38
(
1
), pp.
173
182
.10.2514/2.938
7.
Leonard
,
O.
, and
Van Den Braembussche
,
R.
,
1992
, “
Design Method for Subsonic and Transonic Cascade With Prescribed Mach Number Distribution
,”
ASME J. Turbomach.
,
114
(
3
), pp.
553
560
.10.1115/1.2929179
8.
Demeulenaere
,
A.
,
Leonard
,
O.
, and
Van Den Braembussche
,
R.
,
1997
, “
A Two-Dimensional Navier–Stokes Inverse Solver for Compressor and Turbine Blade Design
,”
J. Power Energy
,
211
(
4
), pp.
299
307
.10.1243/0957650971537204
9.
Coello
,
C.
,
2000
, “
An Updated Survey of GA-Based Multiobjective Optimization Techniques
,”
ACM Comput. Surv.
,
32
(
2
), pp.
109
143
.10.1145/358923.358929
10.
Verstraete
,
T.
,
Alsalihi
,
Z.
, and
Van Den Braembussche
,
R.
,
2010
, “
Multidisciplinary Optimization of a Radial Compressor for Microgas Turbine Applications
,”
ASME J. Turbomach.
,
132
(
2
), p.
031004
.10.1115/1.3144162
11.
Pasquale
,
D.
,
Ghidoni
,
A.
, and
Rebay
,
S.
,
2013
, “
Shape Optimization of an Organic Rankine Cycle Radial Turbine Nozzle
,”
ASME J. Eng. Gas Turbines Power
,
135
(
4
), p.
042308
.10.1115/1.4023118
12.
Lee
,
S. Y.
, and
Kim
,
K. Y.
,
2000
, “
Design Optimization of Axial Flow Compressor Blades With Three-Dimensional Navier–Stokes Solver
,”
KSME Int. J.
,
14
(
9
), pp.
1005
1012
.
13.
Oyama
,
A.
,
2004
, “
Transonic Axial-Flow Blade Optimization: Evolutionary Algorithms/Three-Dimensional Navier–Stokes Solver
,”
J. Propul. Power
,
20
(
4
), pp.
612
619
.10.2514/1.2290
14.
Chen
,
N.
,
Zhang
,
H.
,
Huang
,
W.
, and
Xu
,
Y.
,
2005
, “
Study on Aerodynamic Design Optimization of Turbomachinery Blades
,”
J. Therm. Sci.
,
14
(
4
), pp.
298
304
.10.1007/s11630-005-0048-5
15.
Pierret
,
S.
,
Coelho
,
R.
, and
Kato
,
H.
,
2006
, “
Multidisciplinary and Multiple Operating Points Shape Optimization of Three-Dimensional Compressor Blades
,”
Struct. Multidiscip. Optim.
,
33
(
1
), pp.
61
70
.10.1007/s00158-006-0033-y
16.
Peter
,
J.
, and
Dwight
,
R.
,
2010
, “
Numerical Sensitivity Analysis for Aerodynamic Optimization: A Survey of Approaches
,”
Comput. Fluids
,
39
(
3
), pp.
373
391
.10.1016/j.compfluid.2009.09.013
17.
Quoilin
,
S.
,
Broek
,
M. V. D.
,
Declaye
,
S.
,
Dewallef
,
P.
, and
Lemort
,
V.
,
2013
, “
Techno-Economic Survey of Organic Rankine Cycle (ORC) Systems
,”
Renewable Sustainable Energy Rev.
,
22
(
0
), pp.
168
186
.10.1016/j.rser.2013.01.028
18.
Lang
,
W.
,
Almbauer
,
R.
, and
Colonna
,
P.
,
2013
, “
Assessment of Waste Heat Recovery for a Heavy-Duty Truck Engine Using an ORC Turbogenerator
,”
ASME J. Eng. Gas Turbines Power
,
135
(
4
), p.
042313
.10.1115/1.4023123
19.
Pasquale
,
D.
,
Ghidoni
,
A.
, and
Rebay
,
S.
,
2013
, “
Shape Optimization of an Organic Rankine Cycle Radial Turbine Nozzle
,”
ASME J. Eng. Gas Turbines Power
,
135
(
4
), p.
042305
.10.1115/1.4023118
20.
Cinnella
,
P.
, and
Congedo
,
P.
,
2006
, “
GA-Hardness of Dense-Gas Flow Optimization Problems
,” Applied Simulation and Modelling, Rhodes, Greece, June 26–28.
21.
Giles
,
M.
, and
Pierce
,
N.
,
2000
, “
An Introduction to the Adjoint Approach to Design
,” Oxford Computing Laboratory, Oxford, UK.
22.
Giles
,
M.
,
Ghate
,
D.
, and
Duta
,
M.
,
2005
, “
Using Automatic Differentiation for Adjoint CFD Code Development
,” Oxford Computing Laboratory, Oxford, UK.
23.
Carpentieri
,
G.
,
Koren
,
B.
, and
van Tooren
,
M.
,
2007
, “
Adjoint-Based Aerodynamic Shape Optimization on Unstructured Meshes
,”
J. Comput. Phys.
224
(
1
), pp.
267
287
.10.1016/j.jcp.2007.02.011
24.
Hoschek
,
J.
,
Lasser
,
D.
, and
Schumaker
,
L.
,
1996
,
Fundamentals of Computer Aided Geometric Design
,
A.K. Peters
,
Natick, MA
.
25.
Farin
,
G.
,
2002
,
Curves and Surfaces for CAGD: A Practical Guide
,
5th ed.
,
Academic Press
,
Waltham, MA
.
26.
de Boer
,
A.
,
van der Schoot
,
M.
, and
Bijl
,
H.
,
2007
, “
Mesh Deformation Based on Radial Basis Function Interpolation
,”
Comput. Struct.
,
85
(
11
), pp.
784
795
.10.1016/j.compstruc.2007.01.013
27.
Colonna
,
P.
, and
Rebay
,
S.
,
2004
, “
Numerical Simulation of Dense Gas Flows on Unstructured Grids With an Implicit High Resolution Upwind Euler Solver
,”
Int. J. Numer. Methods Fluids
,
46
(
7
), pp.
735
765
.10.1002/fld.762
28.
Harinck
,
J.
,
Colonna
,
P.
,
Guardone
,
A.
, and
Rebay
,
S.
,
2010
, “
Influence of Thermodynamic Models in 2D Flow Simulations of Turboexpanders
,”
ASME J. Turbomach.
,
132
(
1
), p.
011001
.10.1115/1.3192146
29.
Pecnik
,
R.
,
Rinaldi
,
E.
, and
Colonna
,
P.
,
2012
, “
Computational Fluid Dynamics of a Radial Compressor Operating With Supercritical CO2
,”
ASME J. Eng. Gas Turbines Power
,
134
(
12
), p.
122301
.10.1115/1.4007196
30.
Boger
,
A.
,
2001
, “
Efficient Method for Calculating Wall Proximity
,”
AIAA J.
,
39
(
12
), pp.
2404
2406
.10.2514/2.1251
31.
Pini
,
M.
,
Spinelli
,
A.
,
Persico
,
G.
, and
Rebay
,
S.
,
2014
, “
Consistent Look-Up Table Interpolation Method for Real-Gas Flow Simulations
,”
Comput. Fluids
(in press).
32.
Saad
,
Y.
, and
Schultz
,
M.
,
1986
, “
GMRES: A Generalized Minimal Residual Algorithm for Solving Nonsymmetric Linear Systems
,”
SIAM J. Sci. Stat. Comput.
,
7
(
3
), pp.
856
869
10.1137/0907058.
33.
Hascoet
,
L.
, and
Pascual
,
V.
,
2004
,
Tapenade 2.1 User's Guide
, Unité de recherche INRIA Sophia Antipolis, Cedex, France.
34.
Rinaldi
,
E.
,
Pecnik
,
R.
, and
Colonna
,
P.
,
2014
, “
Implicit Schemes for Compressible Flows of Dense Gases
,”
J. Comput. Phys.
(in press).
35.
Ghidoni
,
A.
,
Pelizzari
,
E.
,
Rebay
,
S.
, and
Selmin
,
V.
,
2006
, “
3D Anisotropic Unstructured Grid Generation
,”
Int. J. Numer. Methods Fluids
,
51
(
9–10
), pp.
1097
1115
.10.1002/fld.1151
36.
Jameson
,
A.
, and
Kim
,
S.
,
2003
, “
Reduction of the Adjoint Gradient Formula in the Continuous Limit
,”
AIAA
Paper No. 2003-0040. 10.2514/6.2003-0040
37.
Kim
,
S.
,
Hosseini
,
K.
,
Leoviriyakit
,
K.
, and
Jameson
,
A.
,
2005
, “
Enhancement of the Adjoint Design Methods Via Optimization of Adjoint Parameters
,”
AIAA
Paper No. 2005-0448. 10.2514/6.2005-0448
38.
Colonna
,
P.
,
Harinck
,
J.
,
Rebay
,
S.
, and
Guardone
,
A.
,
2008
, “
Real-Gas Effects in Organic Rankine Cycle Turbine Nozzles
,”
J. Propul. Power
,
24
(
2
), pp.
282
294
.10.2514/1.29718
39.
Rebay
,
S.
,
Colonna
,
P.
,
Pasquale
,
D.
, and
Ghidoni
,
A.
,
2009
, “
Simulation of the Turbulent Dense Gas Flow Through The Nozzle of an Organic Rankine Cycle Turbine
,”
8th European Conference on Turbomachinery, Fluid Dynamics and Thermodynamics
,
Graz, Austria
, Mar. 23–27, pp.
1137
1148
.
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